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Why do diamonds sparkle? Why is the shower the best place to sing? Where is the 4th dimension in Dali's paintings? If you don't know the answers to these questions, or perhaps
didn't even know you didn't know these interesting facts, then John D. Barrow's book will help. Its title, 100 essential things you didn't know you didn't know about maths and the arts, explains what's in it and you can a taster for Barrow's writing in thisPlus article. You can be sure it's good though, after all this is Barrow's 22nd book!

If you've got maths friends then you might have noticed that a large subset of them are also Simpsons fans. And there's a good reason for that: as Simon Singh's book The Simpsons and their mathematical secrets explains, a large number of Simpsons writers are mathematicians. The book explores the maths that has been hidden (and not so hidden) throughout the series and will make a great present for maths/Simpsons nerds.

In June this year five mathematicians turned into millionaires when they were awarded the Breakthrough Prize set up by Facebook founder Mark Zuckerberg and internet entrepreneur Yuri Milner. Simon Donaldson, Maxim Kontsevich, Jacob Lurie, Terry Tao and Richard Taylor each received £1.8m. That's more than the £930,000 that's awarded for the Nobel prize and should make up for the fact neither of the five is likely to ever to receive a Nobel: there isn't one for mathematics.

Back in November the five winners gave a lectures at the Breakthrough Prize Symposium, which have been filmed for you to watch. We especially recommend Terence Tao's talk on collaboration in maths (mathematicians seem to be getting more sociable):

Richard Taylor's talk about some beautiful number theory (does get a little technical later but starts very accessibly):

And Jacob Lurie's talk on analogy and abstractions in maths, involving some impossible shapes:

Time travel anyone? Human-eating aliens? Exploding Suns? If that sounds good to you, then Keith Mansfield's series of three Johnny Mackintosh books
is what you need. And it doesn't matter how old you are. The hero of the books is thirteen-year-old Johnny Mackintosh, who is busy with football and school work when he picks up signals from extraterrestrials and is catapulted into an adventure that culminates in a mission to save the Earth. Which means that those books are great for anyone aged from about 10 upwards with a taste for adventure and sic-fi — and an interest in mind-bending concepts from real science, which are woven into the plot from the start.

Maths and science related books for children that are fun as well us instructive are rare, so we'd like to mention another couple that have gone down very well with our own offspring. Richard Schwartz's Really big numbers and You can count on monsters are beautifully illustrated books that give kids from around 6 to 8 and intuitive understanding of our number system and prime numbers (respectively). So there's no excuse not to give some maths to your younger friends and family this Christmas!

What's your favourite number? If it's 7 then you're not alone. Alex Bellos, one of our favourite maths authors, conducted a survey this year to find the world's favourite number. After polling more than 30,000 people from around the world, he found that the winner was ... 7!

The square root of half the number of a swarm of bees is gone to a shrub of jasmine; and so are eight-ninth of the whole swarm: a female is buzzing to one remaining male that is humming within a lotus flower in which he is confined, having been allured to it by its fragrance at night. Say, lovely woman, the number of bees.

If only all maths questions could be that poetic! We haven't made this one up, rather it comes from the twelfth-century book Lilavati ("The beautiful"), written by the Indian mathematician Bhaskara. According to legend, Bhaskara had a beautiful daughter whose horoscope predicted she was to remain unmarried and childless. Not wanting to do without grandchildren, Bhaskara decided to defy destiny and built a water clock to determine an "auspicious moment" at which his daughter was to get married. But the girl couldn't resist a peak at the clock — a pearl from her dress fell into it and clogged up the hole, so the auspicious moment could never come. To console her, Bhaskara wrote her a maths book (lucky girl!) and named it after her: Lilavati. It's a beautiful book which would make a great present for anyone who likes maths and history.